Gravity and Strings

3.4 The Fierz–Pauli theory in a curved background 109
isometry group and, in general, there will be no obvious generalizations of these concepts
that work in all cases.
Instead of proceeding case by case trying to give definitions of the mass of a field, we
are going to adopt a general point of view and give a characterization of the masslessness
of a field. The main observation is that massless fields have, as a rule, fewer degrees of
freedom (DOF) than do massive fields, the extra DOF being removed by gauge symmetries
that appear when the mass parameters are set to zero. At the beginning of this chapter
we studied two cases in Minkowski spacetime: a massive vector field has d − 1 DOF and
no gauge symmetries. When we set the mass parameter to zero, the theory has a gauge
symmetry and we can remove one more DOF (a total of two) so there are only the d − 2
DOF of a massless vector. In the spin-2 case, in the presence of mass the field describes
(d − 2)(d + 1)/2 DOF. When we switch off the mass parameter, there appears a gauge
symmetry that allows us to remove d − 1DOF more (a total of 2d) and we are left with the
d(d − 3)/2 DOF of a massless spin-2 particle.
In conclusion, we are going to characterize masslessness by the occurrence of new gauge
symmetries that appear when we switch off the mass parameter.
We have obtained a generalization of the Fierz–Pauli theory to curved backgrounds given
by the action Eq. (3.303) and equation of motion Eq. (3.296) (with vanishing r.h.s.). In this
theory there are terms proportional to the cosmological constant that have the form of
mass terms. To see whether they really are mass terms according to our definition, we look
for gauge symmetries. The obvious candidate is the linearization of the invariance under
GCTs that generalizes Eq. (3.95) to curved backgrounds:
δɛhµν =−2 ¯∇(µɛν), (3.307)
Let us first check the invariance of the action under these transformations. First we vary
the action as usual. We obtain two types of terms: ¯∇h ¯∇ 2ɛ and h ¯∇ɛ (these arise from
the variation of the “mass terms”). We want to move all the derivatives so they act over ɛ.
Thus, we integrate by parts all the terms of the first kind, obtaining h ¯∇ 3ɛ-type terms and
a total derivative. These terms can be combined into terms of the forms h ¯∇[ ¯∇, ¯∇]ɛ and
h[ ¯∇, ¯∇] ¯∇ɛ. Then, the commutators of covariant derivatives can be replaced by curvature
terms using the Ricci identity and all these terms become terms of the type h ¯R ¯∇ɛ and
h ¯∇ ¯Rɛ. The first cancel out, upon use of the vacuum cosmological Einstein equation for the
background metric ¯Rµν = ¯gµν, the h ¯∇ɛ terms. The second cancel out upon use of the
background Bianchi identity ¯∇[µ ¯Rνρ]σ λ = 0and we are left with the total derivative:
δɛ S = 1
χ 2
d d x
|¯g| ¯∇µ
1
2hρσ
4 ¯∇ [µ ¯∇ ρ] ɛ σ − 2 ¯∇ ρ ¯∇ σ ɛ µ
+¯g ρσ ¯∇ 2 ɛ µ − ¯∇λ ¯∇ µ ɛ λ + 2 ¯g µρ ¯∇ σ ¯∇λɛ λ − 2 ¯g ρσ ¯∇ µ ¯∇λɛ λ
≡ d d x |¯g| ¯∇µs µ (ɛ). (3.308)
The Fierz–Pauli equation of motion Eq. (3.296) is, therefore, invariant for the backgrounds
considered. The proof makes crucial use of the Einstein equation satisfied by the
background metric. As we remarked in the introduction to this section, in general backgrounds
there is no way to construct a gauge-invariant theory by adding curvature terms
[47].